Mathematical Research Letters

Volume 17 (2010)

Number 6

Pseudo-Riemannian geometry calibrates optimal transportation

Pages: 1183 – 1197



Young-Heon Kim (University of British Columbia, Vancouver BC Canada and Institute for Advanced Study, Princeton NJ USA)

Robert J. McCann (University of Toronto, Toronto, Ontario, Canada)

Micah Warren (Princeton University, Princeton NJ, USA)


Given a transportation cost $c:M\times\bar{M}\rightarrow\mathbf{R}$, optimal maps minimize the total cost of moving masses from $M$ to $\bar{M}$. We find, explicitly, a pseudo-metric and a calibration form on $M\times\bar{M}$ such that the graph of an optimal map is a calibrated maximal submanifold, and hence has zero mean curvature. We define the mass of space-like currents in spaces with indefinite metrics.

Full Text (PDF format)